DENVER — The fate of one Colorado mass murderer has become entwined with that of another as Colorado Gov. John Hickenlooper struggles with whether to save the life of a notorious death row inmate.

Mr. Hickenlooper is considering a petition to grant clemency for 38-year-old Nathan Dunlap, convicted by a jury in 1996 of killing four employees at a Chuck E. Cheese restaurant in Aurora. The execution, which would be the first in Colorado since 1997, has been set for the week of Aug. 18.

Complicating the process is that any decision the governor makes could have a ripple effect on the case of James Eagan Holmes, 25, the suspect in the highly publicized Aurora theater shooting. The July 20 massacre left 12 dead and 58 injured.

Arapahoe County District Attorney George Brauchler has announced that he will seek the death penalty for Mr. Holmes. If Dunlap’s sentence is commuted to life in prison, however, critics say it would inevitably boost Mr. Holmes‘ chances of avoiding execution.

“The natural question that would flow to John Hickenlooper if he commutes the sentence of Nathan Dunlap is, ‘Why aren’t you doing that for everyone on death row or everyone facing the death penalty?’” said former Denver deputy district attorney Craig Silverman.

Leading the campaign against clemency is Mr. Brauchler, who was still in law school at the time of the Chuck E. Cheese shooting but was elected in November as district attorney for the 18th Judicial District. Both the Dunlap and Holmes cases are being handled by that office.

Mr. Brauchler declined to discuss the ongoing Holmes prosecution, but said that a clemency grant would have consequences that extend far beyond the Dunlap case. There are currently two inmates other than Dunlap on the state’s death row.

“It would be precedential. I don’t remember any governor overturning a death-penalty conviction in Colorado,” said Mr. Brauchler. “I think the anti-death penalty people know this, and that’s why they’re pushing so hard for clemency.”

Indeed, Mr. Hickenlooper is under intense pressure to spare Dunlap’s life from capital-punishment foes as well as members of his own party. His chief of staff, Roxane White, posted messages on Twitter earlier this month expressing her opposition to the Dunlap execution.

Meanwhile, Mr. Hickenlooper infuriated fellow Democrats in March when he snuffed a bill that would have banned capital punishment in Colorado. Granting clemency to Dunlap would help mend fences with his critics on the left, say analysts.

“I think he wants in the worst possible way to grant clemency to Dunlap,” said former Colorado Republican Party chairman Dick Wadhams. “I don’t think the chief of staff goes out there on her own with a view that’s inconsistent with the governor’s viewpoint. That’s just not done.”

In an interview last week with radio host Mike Rosen, the governor said he had spent the last several months meeting with dozens of stakeholders, including prosecutors, defense attorneys and family members of those killed.

“We’ve been working on this for a year, and working very focused for last 41/2 months, so I’ve met now with dozens and dozens and dozens, 60 to 70 family members of victims, not just of this tragedy but other capital punishment cases,” Mr. Hickenlooper said on Mr. Rosen’s KOA-AM show.

When will he decide? “I’m not setting a timeline for myself, but there is an urgency to get this done in the coming weeks,” he said.

In their May 6 clemency petition, Dunlap’s attorneys attest that their client suffers from bipolar disorder, which was undiagnosed at the time of the 1996 trial, and that he was in the throes of a manic episode at the time of the shooting.

The mental health of Mr. Holmes, who showed up at the first several hearings with a blank stare and orange-dyed hair, is likely to be an issue raised by his defense.

Dunlap’s attorneys have also released a video and note from Dunlap in which he apologizes for “the pain and suffering I’ve caused the victims’ families and friends I wish there was something more that I could do to relieve any pain.”

Mr. Brauchler has countered by releasing letters from victims’ relatives, jurors and a former friend of Dunlap’s urging the governor to respect the jury’s decision. Several of those, including Bobby Stephens, the lone shooting survivor, have appeared recently on Denver television talking about the damage inflicted by Dunlap.

Dunlap was 19 when he burst into the restaurant after closing time Dec. 14, 1993, shooting five former co-workers after he was fired. He told authorities afterward that killing them was “better than sex.”

Sandi Rogers, whose 17-year-old son Benjamin Grant was killed in the massacre, said that she and other victims’ family members met with the governor May 3. Her message for him: Do nothing.

“What I’ve been saying is that there needs to be more about Ben [and the other victims], because if their names were mentioned every time Nathan Dunlap’s name was mentioned, people would realize this isn’t about Nathan Dunlap,” said Mrs. Rogers. “This is about those four people. That’s where the focus needs to be.”